See UVs for beveled curve in UV/Image Editor
Try this:
https://www.deviantart.com/vickym72/art/Blender-Empty-Decal-Tutorial-207942535

It's essentially a tutorial for manipulating decals using an empty, but you can use it for the curve as well. (Credits to: VickyM72)

Hope this helps :)

EDIT: Here's how to do it in cycles --> https://www.youtube.com/watch?v=dnmovLzKU44&amp;ab_channel=KarstenH%C3%B6hne (Credits to: Karsten Höhne
)

EDIT 2: try this:

1> go to the curve tab and open Texture space

2> click Use UV for mapping

3> Click match texture space

4> Go to material node editor

5> Add your image texture

6> Add Texture coordinate node and hookup the vector of the image texture to generated

7> use the Location and Size values to manually manipulate your texture

EDIT 3: I made a video: https://youtu.be/bx1FdoG2xo8

Hope that works :)